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Uncertainty in Mechanical Engineering

Proceedings of the 4th International Conference on Uncertainty in Mechanical Engineering (ICUME 2021), June 7–8, 2021

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About this book

This open access book reports on methods and technologies to describe, evaluate and control uncertainty in mechanical engineering applications. It brings together contributions by engineers, mathematicians and legal experts, offering a multidisciplinary perspective on the main issues affecting uncertainty throughout the complete system lifetime, which includes process and product planning, development, production and usage. The book is based on the proceedings of the 4th International Conference on Uncertainty in Mechanical Engineering (ICUME 2021), organized by the Collaborative Research Center (CRC) 805 of the TU Darmstadt, and held online on June 7–8, 2021. All in all, it offers a timely resource for researchers, graduate students and practitioners in the field of mechanical engineering, production engineering and engineering optimization.

Table of Contents

Frontmatter

Mastering Uncertainty by Digitalization

Frontmatter

Open Access

Ontology-Based Calculation of Complexity Metrics for Components in CAD Systems
Abstract
The high complexity of assemblies and components in Computer-Aided Design (CAD) leads to a high effort in the maintenance of the models and increases the time required for adjustments. Metrics indicating the complexity of a CAD Model can help to reduce it by showing the results of changes. This paper describes a concept to calculate metrics aiming to describe the extent of complexity of components in CAD systems based on an ontology-based representation in a first step. The representation is initially generated from CAD models using an automated process. This includes both a boundary representation and the history of the feature-based design. Thus, the design strategy also contributes to measuring the complexity of the component so that the same shape can lead to different complexity metrics. Semantic rules are applied to find patterns of the design and to identify and evaluate various strategies. Different metrics are proposed to indicate the particular influence factors of complexity and a single measure for the overall complexity. Furthermore, the influencing factors can also be used to allow the designer to see how to reduce the complexity of the component or assembly.
Moritz Weber, Reiner Anderl

Open Access

Towards CAD-Based Mathematical Optimization for Additive Manufacturing – Designing Forming Tools for Tool-Bound Bending
Abstract
The trend towards flexible, agile, and resource-efficient production systems requires a continuous development of processes as well as of tools in the area of forming technology. To create load-adjusted and weight-optimized tool structures, we present an overview of a new algorithm-driven design optimization workflow based on mixed-integer linear programming. Loads and boundary conditions for the mathematical optimization are taken from numerical simulations. They are transformed into time-independent point loads generating physical uncertainty in the parameters of the optimization model. CAD-based mathematical optimization is used for topology optimization and geometry generation of the truss-like structure. Finite element simulations are performed to validate the structural strength and to optimize the shape of lattice nodes to reduce mechanical stress peaks. Our algorithm-driven design optimization workflow takes full advantage of the geometrical freedom of additive manufacturing by considering geometry-based manufacturing constraints. Depending on the additive manufacturing process, we use lower and upper bounds on the diameter of the members of a truss and the associated yield strengths. An additively manufactured flexible blank holder demonstrates the algorithm-driven topology design optimization.
Christian Reintjes, Jonas Reuter, Michael Hartisch, Ulf Lorenz, Bernd Engel

Open Access

Development of an Annotation Schema for the Identification of Semantic Uncertainty in DIN Standards
Abstract
This paper presents the results of a pilot study carried out in cooperation between Linguistics and Mechanical Engineering, funded by the collaborative research centre (CRC) 805 “Beherrschung von Unsicherheit in lasttragenden Systemen des Maschinenbaus”. Our goal is to help improve norm compliant product development and engineering design by focusing on ambiguous language use in norm texts (= “semantic uncertainty”). Depending on the country and product under development, industry standards may be legally binding. Thus, standards play a vital role in reducing uncertainty for manufacturers and engineers by providing requirements for product development and engineering design. However, uncertainty is introduced by the standards themselves in various forms, the most notable of which are the use of underspecified concepts, modal verbs like should, and references to texts which contain semantically uncertain parts. If conformity to standards is to be ensured, the person using the standards must interpret them and document the interpretation. In order to support users in these tasks, we
1.
developed an annotation schema which allows the identification and classification of semantically uncertain segments of standards,
 
2.
used the schema to create a taxonomy of semantic uncertainty in standards,
 
3.
developed a proof-of-concept information system.
 
The results of this project can be used as a starting point for automated annotation. The information system alerts users to semantically uncertain segments of standards, provides background information, and allows them to document their decisions how to handle the semantically uncertain parts.
Jörn Stegmeier, Jakob Hartig, Michaela Leštáková, Kevin Logan, Sabine Bartsch, Andrea Rapp, Peter F. Pelz

Open Access

Mastering Model Uncertainty by Transfer from Virtual to Real System
Abstract
Two chassis components were developed at the Technische Universität Darmstadt that are used to isolate the body and to reduce wheel load fluctuation.
The frequency responses of the components were identified with a stochastic foot point excitation in a hardware-in-the-loop (HiL) simulation environment at the hydropulser. The modelling of the transmission behaviour influence of the testing machine on the frequency response was approximately represented with a time delay of \(10\,\mathrm {ms}\) in the frequency range up to \(25\,\mathrm {Hz}\). This is considered by a Padé approximation. It can be seen that the dynamics of the testing machine have an influence on the wheel load fluctuation and the body acceleration, especially in the natural frequency of the unsprung mass. Therefor, the HiL stability is analysed by mapping the poles of the system in the complex plane, influenced by the time delay and virtual damping.
This paper presents the transfer from virtual to real quarter car to quantify the model uncertainty of the component, since the time delay impact does not occur in the real quarter car test rig. The base point excitation directly is provided by the testing machine and not like in the case of the HiL test rig, the compression of the spring damper calculated in the real-time simulation.
Nicolas Brötz, Manuel Rexer, Peter F. Pelz

Resilience

Frontmatter

Open Access

Potentials and Challenges of Resilience as a Paradigm for Designing Technical Systems
Abstract
The resilience paradigm constitutes that systems can overcome arbitrary system failures and recover quickly. This paradigm has already been applied successfully in multiple disciplines outside the engineering domain. For the development and design of engineering systems the realization of this resilience concept is more challenging and often leads to confusion, because technical systems are characterized by a lower intrinsic complexity compared to, e.g., socio-technical systems. The transfer of the resilience paradigm to technical systems though also offers high potential for the engineering domain. We present results from four-year research on transferring the resilience paradigm to the engineering domain based on mechanical engineering systems and summarize relevant design approaches to quantify the potentials of this paradigm. Furthermore, we present important challenges we faced while transferring this paradigm and present the lessons learned from this interdisciplinary research.
Philipp Leise, Pia Niessen, Fiona Schulte, Ingo Dietrich, Eckhard Kirchner, Peter F. Pelz

Open Access

Modelling of Resilient Coping Strategies within the Framework of the Resilience Design Methodology for Load-Carrying Systems in Mechanical Engineering
Abstract
During the development of load-carrying systems uncertainty caused by nescience can be handled applying resilience design. With this systematic approach, in addition to robust design, resilient system properties can be achieved. The resilience design methodology comprises new and extended models and methods. The central aspect of resilient properties is an adaptivity of the system. The procedure for resilience design starts with choosing a ‘general coping strategy’ appropriate for the design task. Based on this, a more detailed ‘system coping strategy’ is developed. This concrete strategy is based on the resilience functions responding, monitoring, anticipating and learning. The coping strategies always contain the function ‘responding’ because it represents the actual adaption of the system. The central, most abstract synthesis model for developing robust and resilient systems is the functional structure model. In this model the system functions and their interconnection by signals, material and energy flows are depicted. However, the realisation of resilience properties requires additional signals and flows. Hitherto, the functional structure for robust systems is static, whereas adaptivity requires flexible control of functions and flows. Therefore, an extension of the functional structure model is proposed to be able to depict the resilient system coping strategy and adaptivity. Within the resilient system the coping strategy is depicted by adaption functions based on the four resilience functions. Via an introduced interface and an enabler-structure the adaption functions are connected to the robust functional structure. The application of the proposed extension is illustrated by the example of a by-wire car brake system.
Fiona Schulte, Hermann Kloberdanz, Eckhard Kirchner

Open Access

Validation of an Optimized Resilient Water Supply System
Abstract
Component failures within water supply systems can lead to significant performance losses. One way to address these losses is the explicit anticipation of failures within the design process. We consider a water supply system for high-rise buildings, where pump failures are the most likely failure scenarios. We explicitly consider these failures within an early design stage which leads to a more resilient system, i.e., a system which is able to operate under a predefined number of arbitrary pump failures. We use a mathematical optimization approach to compute such a resilient design. This is based on a multi-stage model for topology optimization, which can be described by a system of nonlinear inequalities and integrality constraints. Such a model has to be both computationally tractable and to represent the real-world system accurately. We therefore validate the algorithmic solutions using experiments on a scaled test rig for high-rise buildings. The test rig allows for an arbitrary connection of pumps to reproduce scaled versions of booster station designs for high-rise buildings. We experimentally verify the applicability of the presented optimization model and that the proposed resilience properties are also fulfilled in real systems.
Tim M. Müller, Andreas Schmitt, Philipp Leise, Tobias Meck, Lena C. Altherr, Peter F. Pelz, Marc E. Pfetsch

Open Access

Comparability of Water Infrastructure Resilience of Different Urban Structures
Abstract
Urban water distribution systems (WDS) ensure the demand-driven supply of a city at multiple ends. Well-being of the population as well as multiple economic sectors depend on its viability and thereby classify it as a critical infrastructure. Therefore, its behavior when exposed to changes is of interest to water suppliers as well as local decision-makers. It can be determined by resilience metrics, assessing the capability to meet and recover its functioning when exposed to disturbances. These disturbances can occur in form of changes in the water availability, the WDS topology, or the water demand pattern. Since networks as WDS are studied by graph theory, also different graph-theoretical resilience metrics were derived. In this work a well-established topology-based resilience metric is adapted and deployed to assess the present resilience of the urban main-line WDS of the German city of Darmstadt as well as of a suburb in the Rhine-Main region. Thereby, the intercomparability of the resilience for the different urban structures were of interest. Based on this analysis the comparability of different urban main-line WDS regarding their resilience is facilitated. Additionally, the conducted approach to allow for the comparability of absolute resilience values of urban structures of varying size can be applied to different resilience metrics as well as technical systems.
Imke-Sophie Lorenz, Kevin Pouls, Peter F. Pelz

Uncertainty in Production

Frontmatter

Open Access

Dealing with Uncertainties in Fatigue Strength Using Deep Rolling
Abstract
Mechanical properties inherently possess uncertainties. Among these properties, fatigue behavior data generally shows significant scatter which introduces a challenge in the safe design of dynamically loaded components. These uncertainties in fatigue behavior are mainly results of factors related to surface state including: Roughness, tensile residual stresses, scratches and notches at surface. Therefore, controlling these parameters allows one to increase fatigue strength and reduce scatter and uncertainties in fatigue behavior. Mechanical surface treatments are applied on parts to increase fatigue strength via introducing compressive residual stresses and work-hardening at surface. Two of the most common among these treatments are shot peening and deep rolling. Shot peening has found many applications in industry because of its flexibility. However, it introduces irregularities at the surface and may increase roughness which causes uncertainties in the fatigue behavior data; especially for low-medium strength materials. Unlike shot peening, deep rolling reduces surface roughness. Therefore, it has the capability to reduce uncertainty in the fatigue behavior. To this date, rolling direction of deep rolling was selected as tangential direction to turning direction for axisymmetric parts. Nonetheless, the authors believe that the rolling direction has an apparent effect on the fatigue behavior. In this study, longitudinal direction was also applied for deep rolling operation and the results of these two direction applications on the EN-AW-6082 aluminum alloy were investigated. It was shown that, longitudinal rolling had yielded less scatter and uncertainty in the fatigue behavior than the tangential rolling together with the higher fatigue strength.
Berkay Yüksel, Mehmet Okan Görtan

Open Access

Investigation on Tool Deflection During Tapping
Abstract
Tapping is a challenging process at the end of the value chain. Hence, tool failure is associated with rejected components or expensive rework. For modelling the tapping process we choose a mechanistic approach. In the present work, we focus on the tool model, which describes the deflection and inclination of the tool as a result of the radial forces during tapping. Since radial forces always occur during tapping due to the uneven load distribution on the individual teeth, the tool model represents an essential part of the entire closed-loop model. Especially in the entry phase of the tap, when the guidance within the already cut thread is not yet given, radial forces can lead to deflection of the tool. Therefore, the effects of geometric uncertainty in the thread geometry are experimentally investigated, using optical surface measurement to evaluate the position of the thread relative to the pre-drilled bore. Based on the findings, the tool deflection during tapping is mapped using a cylindrical cantilever beam model, which is calibrated using experimental data. The model is then validated and the implementation within an existing model framework is described.
Felix Geßner, Matthias Weigold, Eberhard Abele

Open Access

How to Predict the Product Reliability Confidently and Fast with a Minimum Number of Samples in the Wöhler Test
Abstract
To accurately estimate and predict the (product) lifetime, a large sample size is mandatory, especially for new and unknown materials. The realization of such a sample size is rarely feasible for reasons of cost and capacity. The prior knowledge must be systematically and consistently used to be able to predict the lifetime accurately. By using the example of Wöhler test, it will be shown that the lifetime prediction with a minimum number of specimen and test time can be successful, when taking the prior knowledge into account.
Jens Mischko, Stefan Einbock, Rainer Wagener

Open Access

Tuning and Emulation of Mechanical Characteristics – Tunable Mounts and a Mechanical Hardware-in-the-Loop Approach for More Efficient Research and Testing
Abstract
Numerical simulations offer a wide range of benefits, therefore they are widely used in research and development. One of the biggest benefits is the possibility of automated parameter variation. This allow testing different scenarios in a very short period of time. Nevertheless, physical experiments in the laboratory or on a test rig are still necessary and will still be necessary in the future. The physical experiments offer benefits e.g. for very complex and/or nonlinear systems and are needed for the validation of numerical models.
Fraunhofer LBF has developed hardware solutions to bring the benefit of rapid and automated parameter variation to experimental environments. These solutions allow the tuning and emulation of the mechanical properties, like stiffness, damping and eigenfrequencies of structures.
The work presents two approaches: First a stiffness tunable mount, which has been used in laboratory tests in the field of semi-active load path redistribution. It allowed the researcher to test the semi-active system under different mechanical boundary conditions in a short period of time. Second, a mechanical Hardware-in-the-loop (mHIL) approach for the NVH development of vehicles components is presented. Here a mHIL-system is used to emulate the mechanical characteristics of a vehicle’s body in white in a wide frequency range. This allows the experimental NVH optimization of vehicle components under realistic boundary conditions, without actually needing a (prototype) body in white.
Jonathan Millitzer, Jan Hansmann, Giovanni Lapiccirella, Christoph Tamm, Sven Herold

Open Access

Identifying and Mastering Legal Uncertainty Concerning Autonomous Systems
Abstract
The level of uncertainty concerning the use of autonomous systems is still very high. This also poses a liability risk for manufactures, which can impede the pace of innovation. Legal uncertainty also contributes to this factor. This paper will discuss existing legal uncertainty. The identified uncertainty can stem from different sources. Categorizing these sources will be our first step when trying to master legal uncertainty. On the basis of these categories, we will be able to evaluate where the focus for mastering legal uncertainty should lie. This approach promises to identify true legal uncertainty, which can only be mastered by new legislation, and separate it from other forms of legal uncertainty which can stem from unclear legal guidelines or uncertainty regarding the application and scope of existing rules and guidelines. Mastering the latter could be possible by specifying said existing rules and guidelines or even by clarifying the scope of their application, a much less drastic solution. Establishing how to deal with different categories of legal uncertainty will then contribute to minimizing liability risks for manufacturers.
Laura Joggerst, Janine Wendt

Uncertainty Quantification

Frontmatter

Open Access

Identification of Imprecision in Data Using -Contamination Advanced Uncertainty Model
Abstract
One of the importance of the contamination uncertainty model is to consider in-determinism in the uncertainty. We consider this advanced property and develop two methods. These methods identify if there is imprecision in a given model or data. In the first approach, we build two different—a probability distribution and an interval—models for a test function f via given data/model. Then, we identify the level of imprecision by assessing, so-called model trust, \(\epsilon \in (0,1)\) in the contamination model whether the weight is higher for the probabilistic/interval model or not. In the second approach, we calculate the lowest and highest previsions for the test function and identify the imprecision interval out of them. We further discuss and show the idea via two simple production and clutch design problems to illustrate our novel results.
Keivan Shariatmadar, Hans Hallez, David Moens

Open Access

Forward vs. Bayesian Inference Parameter Calibration: Two Approaches for Non-deterministic Parameter Calibration of a Beam-Column Model
Abstract
Mathematical models are commonly used to predict the dynamic behavior of mechanical structures or to synthesize controllers for active systems. Calibrating the model parameters to experimental data is crucial to achieve reliable and adequate model predictions. However, the experimental dynamic behavior is uncertain due to variations in component properties, assembly and mounting. Therefore, uncertainty in the model parameters can be considered in a non-deterministic calibration. In this paper, we compare two approaches for a non-deterministic parameter calibration, which both consider uncertainty in the parameters of a beam-column model. The goal is to improve the model prediction of the axial load-dependent lateral dynamic behavior. The investigation is based on a beam-column system subjected to compressive axial loads used for active buckling control. A representative sample of 30 nominally identical beam-column systems characterizes the variations in the experimental lateral axial load-dependent dynamic behavior. First, in a forward parameter calibration approach, the parameters of the beam-column model are calibrated separately for all 30 investigated beam-column systems using a least squares optimization. The uncertainty in the parameters is obtained by assuming normal distributions of the separately calibrated parameters. Second, in a Bayesian inference parameter calibration approach, the parameters are calibrated using the complete sample of experimental data. Posterior distributions of the parameters characterize the uncertain dynamic behavior of the beam-column model. For both non-deterministic parameter calibration approaches, the predicted uncertainty ranges of the axial load-dependent lateral dynamic behavior are compared to the uncertain experimental behavior and the most accurate results are identified.
Maximilian Schaeffner, Christopher M. Gehb, Robert Feldmann, Tobias Melz

Open Access

Surrogate Model-Based Uncertainty Quantification for a Helical Gear Pair
Abstract
Competitive industrial transmission systems must perform most efficiently with reference to complex requirements and conflicting key performance indicators. This design challenge translates into a high-dimensional multi-objective optimization problem that requires complex algorithms and evaluation of computationally expensive simulations to predict physical system behavior and design robustness. Crucial for the design decision-making process is the characterization, ranking, and quantification of relevant sources of uncertainties. However, due to the strict time limits of product development loops, the overall computational burden of uncertainty quantification (UQ) may even drive state-of-the-art parallel computing resources to their limits. Efficient machine learning (ML) tools and techniques emphasizing high-fidelity simulation data-driven training will play a fundamental role in enabling UQ in the early-stage development phase.
This investigation surveys UQ methods with a focus on noise, vibration, and harshness (NVH) characteristics of transmission systems. Quasi-static 3D contact dynamic simulations are performed to evaluate the static transmission error (TE) of meshing gear pairs under different loading and boundary conditions. TE indicates NVH excitation and is typically used as an objective function in the early-stage design process. The limited system size allows large-scale design of experiments (DoE) and enables numerical studies of various UQ sampling and modeling techniques where the design parameters are treated as random variables associated with tolerances from manufacturing and assembly processes. The model accuracy of generalized polynomial chaos expansion (gPC) and Gaussian process regression (GPR) is evaluated and compared. The results of the methods are discussed to conclude efficient and scalable solution procedures for robust design optimization.
Thomas Diestmann, Nils Broedling, Benedict Götz, Tobias Melz

Open Access

Approach to Assess Basic Deterministic Data and Model Form Uncertaint in Passive and Active Vibration Isolation
Abstract
This contribution continues ongoing own research on uncertainty quantification in structural vibration isolation in early design stage by various deterministic and non-deterministic approaches. It takes into account one simple structural dynamic system example throughout the investigation: a one mass oscillator subject to passive and active vibration isolation. In this context, passive means that the vibration isolation only depends on preset inertia, damping, and stiffness properties. Active means that additional controlled forces enhance vibration isolation. The simple system allows a holistic, consistent and transparent look into mathematical modeling, numerical simulation, experimental test and uncertainty quantification for verification and validation. The oscillator represents fundamental structural dynamic behavior of machines, trusses, suspension legs etc. under variable mechanical loading. This contribution assesses basic experimental data and mathematical model form uncertainty in predicting the passive and enhanced vibration isolation after model calibration as the basis for further deterministic and non-deterministic uncertainty quantification measures. The prediction covers six different damping cases, three for passive and three for active configuration. A least squares minimization (LSM) enables calibrating multiple model parameters using different outcomes in time and in frequency domain from experimental observations. Its adequacy strongly depends on varied damping properties, especially in passive configuration.
Roland Platz

Open Access

Reconstructing Stress Resultants in Wind Turbine Towers Based on Strain Measurements
Abstract
Support structures of offshore wind turbines are subject to cyclic stresses generated by different time-variant random loadings such as wind, waves, and currents in combination with the excitation by the rotor. In the design phase, the cyclic demand on wind turbine support structure is calculated and forecasted with semi or fully probabilistic engineering models. In some cases, additional cyclic stresses may be induced by construction deviations, unbalanced rotor masses and structural dynamic phenomena such as, for example, the Sommerfeld effect. Both, the significant uncertainties in the design and a validation of absence of unforeseen adverse dynamic phenomena necessitate the employment of measurement systems on the support structures. The quality of the measurements of the cyclic demand on the support structures depends on (a) the precision of the measurement system consisting of sensors, amplifier and data normalization and (b) algorithms for analyzing and converting data to structural health information. This paper presents the probabilistic modelling and analysis of uncertainties in strain measurements performed for the purposes of reconstructing stress resultants in wind turbine towers. It is shown how the uncertainties in the strain measurements affect the uncertainty in the individual components of the reconstructed forces and moments. The analysis identifies the components of the vector of stress resultants that can be reconstructed with sufficient precision.
Marko Kinne, Ronald Schneider, Sebastian Thöns

Open Access

Mastering Uncertain Operating Conditions in the Development of Complex Machine Elements by Validation Under Dynamic Superimposed Operating Conditions
Abstract
Machine elements produced in large quantities undergo several development cycles and can be adapted from generation to generation. Thus, experiences from real operation can be taken into account in further development. This is not possible for innovative investment goods such as special purpose machines, as these are usually individual items. Therefore, functionality and quality of newly developed components must be assured by previous investigations.
Conventional methods are inadequate at this point, as they cannot represent the actual, complex operating conditions in the later application. A reliable statement about the behavior of the system through a comprehensive validation in laboratory tests under standardized conditions is not achievable in this way due to a multitude of diversified load cases.
In previous work, a method was developed to allow testing of machine elements in the laboratory under detuned operating conditions. For this purpose, disturbance variables are applied to the system using paraffin wax phase change actuators in order to simulate real operation states and to analyze the behavior of the machine element under these conditions. The investigated disturbance variables are fluctuations and asymmetries of the operating load through superimposed temperature gradients. Complex interactions between the machine element and the adjacent components or the overall system can thus be taken into account.
The functionality of the methodology has been developed and briefly demonstrated so far. This paper presents the next level within the development process of the methodology. The necessary components are explained in detail and an AI black box evaluation tool is discussed. This work is based on a test bench that applies dynamically changing states of detuning under superimposed disturbances. Additionally, energy efficiency and performance of the test setup is advanced. As presented, the method opens up the possibility of validating new machine elements in the laboratory under realistic conditions.
Thiemo Germann, Daniel M. Martin, Christian Kubik, Peter Groche

Open Access

On Uncertainty, Decision Values and Innovation
Abstract
This paper contains a description, an alignment and a joint approach for technology readiness development with a three phases support of decision value analyses. The three phases are separated into the decision value forecasting, decision value analysis and the technology value quantification supporting the technological concept formulation and experimental testing, the prototype development and the technology qualification and operation. Decision value forecasting allows technology development guidance by technology performance requirements and the value creation even before the technology development is started. This approach is exemplified with load, damage and resistance information based integrity management of a structure and the ranking of the different strategies. The results can be used to guide a technology screening for matching with performance characteristics in terms of precision, cost and employability. Moreover, the first estimate of value creation of the technology for stakeholders, business models and market evaluation is provided.
Sebastian Thöns, Arifian Agusta Irman, Maria Pina Limongelli

Open Access

Assessment of Model Uncertainty in the Prediction of the Vibroacoustic Behavior of a Rectangular Plate by Means of Bayesian Inference
Abstract
Designing the vibroacoustic properties of thin-walled structures is of particularly high practical relevance in the design of vehicle structures. The vibroacoustic properties of thin-walled structures, e.g., vehicle bodies, are usually designed using finite element models. Additional development effort, e.g., experimental tests, arises if the quality of the model predictions are limited due to inherent model uncertainty. Model uncertainty of finite element models usually occurs in the modeling process due to simplifications of the geometry or boundary conditions. The latter highly affect the vibroacoustic properties of a thin-walled structure. The stiffness of the boundary condition is often assumed to be infinite or zero in the finite element model, which can lead to a discrepancy between the measured and the calculated vibroacoustic behavior. This paper compares two different boundary condition assumptions for the finite element (FE) model of a simply supported rectangular plate in their capability to predict the vibroacoustic behavior. The two different boundary conditions are of increasing complexity in assuming the stiffness. In a first step, a probabilistic model parameter calibration via Bayesian inference for the boundary conditions related parameters for the two FE models is performed. For this purpose, a test stand for simply supported rectangular plates is set up and the experimental data is obtained by measuring the vibrations of the test specimen by means of scanning laser Doppler vibrometry. In a second step, the model uncertainty of the two finite element models is identified. For this purpose, the prediction error of the vibroacoustic behavior is calculated. The prediction error describes the discrepancy between the experimental and the numerical data. Based on the distribution of the prediction error, which is determined from the results of the probabilistic model calibration, the model uncertainty is assessed and the model, which most adequately predicts the vibroacoustic behavior, is identified.
Nikolai Kleinfeller, Christopher M. Gehb, Maximilian Schaeffner, Christian Adams, Tobias Melz

Optimization Under Uncertainty

Frontmatter

Open Access

Detection of Model Uncertainty in the Dynamic Linear-Elastic Model of Vibrations in a Truss
Abstract
Dynamic processes have always been of profound interest for scientists and engineers alike. Often, the mathematical models used to describe and predict time-variant phenomena are uncertain in the sense that governing relations between model parameters, state variables and the time domain are incomplete. In this paper we adopt a recently proposed algorithm for the detection of model uncertainty and apply it to dynamic models. This algorithm combines parameter estimation, optimum experimental design and classical hypothesis testing within a probabilistic frequentist framework. The best setup of an experiment is defined by optimal sensor positions and optimal input configurations which both are the solution of a PDE-constrained optimization problem. The data collected by this optimized experiment then leads to variance-minimal parameter estimates. We develop efficient adjoint-based methods to solve this optimization problem with SQP-type solvers. The crucial test which a model has to pass is conducted over the claimed true values of the model parameters which are estimated from pairwise distinct data sets. For this hypothesis test, we divide the data into k equally-sized parts and follow a k-fold cross-validation procedure. We demonstrate the usefulness of our approach in simulated experiments with a vibrating linear-elastic truss.
Alexander Matei, Stefan Ulbrich

Open Access

Robust Topology Optimization of Truss-Like Space Structures
Abstract
Due to the additional design freedom and manufacturing possibilities of additive manufacturing compared to traditional manufacturing, topology optimization via mathematical optimization gained importance in the initial design of complex high-strength lattice structures. We consider robust topology optimization of truss-like space structures with multiple loading scenarios. A typical dimensioning method is to identify and examine a suspected worst-case scenario using experience and component-specific information and to incorporate a factor of safety to hedge against uncertainty. We present a quantified programming model that allows us to specify expected scenarios without having explicit knowledge about worst-case scenarios, as the resulting optimal structure must withstand all specified scenarios individually. This leads to less human misconduct, higher efficiency and, thus, to cost and time savings in the design process. We present three-dimensional space trusses with minimal volume that are stable for up to 100 loading scenarios. Additionally, the effect of demanding a symmetric structure and explicitly limiting the diameter of truss members in the model is discussed.
Michael Hartisch, Christian Reintjes, Tobias Marx, Ulf Lorenz
Backmatter
Metadata
Title
Uncertainty in Mechanical Engineering
Editors
Prof. Dr. Peter F. Pelz
Prof. Dr. Peter Groche
Copyright Year
2021
Electronic ISBN
978-3-030-77256-7
Print ISBN
978-3-030-77255-0
DOI
https://doi.org/10.1007/978-3-030-77256-7

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